No, adding a GPU will not save you. Your system has, if I guess the right version of your hardware, an AMD E1-1200 1.4 GHz APU powering it. Like most of the E series of APUs, this machine actually already has a significant GPU (Radeon HD 7310) embedded in it. However, your CPU side is severely lacking in performance, with only two cores operating at 1.4ghz each. Because HTML5 video from YouTube can be relatively CPU intensive, I suspect your CPU's performance is at fault, not your GPU's.
The answer is most likely to switch from default HTML5 video encoding to Flash; or possibly to update/switch your GPU drivers so that if you are using GPU-accelerated video encoding already, your GPU will be better utilized. Fglrx may provide superior performance here compared with Gallium; it's definitely something to try.
Unfortunately, your APU is BGA-soldered to your motherboard, so there's no way to just improve your hardware. If upgrading becomes necessary, it would make sense for you to take a look at AMD's socket AM1 or Intel's embedded Celeron/Pentium options since Haswell; either of these approaches would yield a similarly efficient and cheap machine with a considerable increase in performance both on the CPU and GPU side.